Voldemort has taken extreme measures to remain "alive" so he can carry out revenge against Harry and the wizards who defeated his evil plan years ago. One of those measures is consuming Unicorn blood.
Hagrid informed Harry, Ron, and Hermione "Drinking unicorn blood will keep you 'alive' but it takes something from you in the process." Its actually very much like Vampires and Zombies: they're not really "alive" they're undead. The same thing comes from drinking unicorn blood, it renders the person "undead," but doesn't impart true immortality.
The only thing that can reverse the effects of Unicorn blood is the "elixir of Life," which is derived from the Sorcerer's stone, according to Hermione's research. That is why Voldemort must find the Sorcerer's Stone, it is the only way he can obtain the elixir of life and recover from using Unicorn blood to remain undead. Then he can kill Harry and resume his conquest.
The Sorcerer's Stone, or Philosopher's Stone, provides "immortality" of sorts to whoever possess it. (This is actually part of folklore, and Nicholas Flamel, the famous supposed owner of the Sorcerer's Stone, was an actual person). Of course, most anyone would love to live forever, and Voldemort wants it terribly so he can try to take over the world again.
The Sorcerer's stone promises immortality. Voldemort desire immortality and hence he desires the stone. But an even more pressing matter is that at this point in the series, Voldemort does not have a body. The stone could provide him with one and that is very important to him.